FIG. 1 is a conventional amplitude-time diagram of three phases of AC current or voltage, with a 120.degree. relative temporal phase .omega.t, which is often used to describe three-phase systems. The abbreviation AC stands for "alternating current," which in many cases is a misnomer, as the term may refer to either alternating voltage or alternating current, or both. A voltage or current is "alternating" if the instantaneous value dwells periodically at positive values followed by a dwell at a negative value, where the terms "positive" or "negative" refer to direction in an electrical, rather than mechanical, sense. In FIG. 1, the three phases are designated X.sub.a, X.sub.b, and X.sub.c, and may be expressed as X.sub.a =X cos wt, X.sub.b =X cos (wt-2.pi./3), and X.sub.b =X cos (wt-4.pi./3), respectively. As known to those skilled in the art, FIG. 2 is a two-dimensional vector representation of the currents or voltages of FIG. 1, taken at time t=0. In FIG. 2, two orthogonal .alpha. and .beta. components are the fundamental ordinates, and the .alpha. axis corresponds with an "a" axis. Two additional b and c axes lie 120.degree. from the a axis. Voltage or current X.sub.a always lies along the a axis, and its instantaneous amplitude changes with time. Similarly, X.sub.b and X.sub.c always lie along the b and c axes, respectively, and their amplitudes also change with time. At the time illustrated in FIG. 2, which corresponds to time t=0 of FIG. 1, the amplitudes of X.sub.a, X.sub.b and X.sub.c of FIG. 2 are such as to sum together, to correspond to a vector. As time or temporal phase increases in the plots of X.sub.a, X.sub.b, and X.sub.c of FIG. 1, vector rotates counterclockwise (in the direction of the arrow .omega.t) in FIG. 2, tracing out a circle 210. A phase change of 2.pi. results in the tracing of one complete rotation about circle 210. In order to simplify mathematical operations, the values of X.sub.a, X.sub.b, and X.sub.c may be expressed in terms of the corresponding components .alpha. and .beta.. ##EQU1## where ##EQU2## Equations (1) and (2) taken together represent the matrix transformation that converts a vector expressed in terms of a, b, and c into a vector expressed in terms of .alpha. and .beta. components. The above mathematical representation does not take into account common-mode (direct or non-alternating, or alternating) voltage or current which may be associated with the alternating energization. In an AC system, each phase may be associated with a common-mode component, which offsets the alternating component of that phase. The plot of FIG. 2 assumes or represents a situation in which there is no common-mode component. If there is a common-mode component in a balanced three-phase system, it cancels and does not appear in the alpha and beta components. Since the common-mode components cancel, a plot such as that of FIG. 2 would not show them, even if present. However, such a common-mode component may exist, even if it is not represented in the two-dimensional plot of FIG. 2.
In order to represent the direct components, vectorX may be represented by the modified transformation set forth below in relation to equations (3) and (4) ##EQU3## where the elements of the lowest row of the matrix, having value "1/2," represent the common-mode component. The additional X.sub.0 component is known as the zero sequence component. The common-mode component which is represented by the zero-sequence component are often represented by a vector orthogonal to the .alpha.--.beta. plane of FIG. 2. Application of such a vector orthogonal to the .alpha.--.beta. plane results in a three-dimensional system of mutually orthogonal coordinates.